On a relation between the self-linking number and the braid index of closed braids in open books

TL;DR

This paper generalizes the Jones-Kawamuro conjecture, establishing a relation between self-linking number and braid index for certain planar open books, and demonstrates the optimality of this relation through counterexamples.

Contribution

It extends the Jones-Kawamuro conjecture to broader conditions in planar open books and shows the limits of naive generalizations.

Findings

Established a generalized relation between self-linking number and braid index.

Provided counterexamples illustrating the boundaries of the conjecture.

Demonstrated the optimality of the generalized relation.

Abstract

We prove a generalization of the Jones-Kawamuro conjecture that relates the self-linking number and the braid index of closed braids, for planar open books with certain additional conditions and modifications. We show that our result is optimal in some sense by giving several counter examples for naive generalizations of the Jones-Kawamuro conjecture.